Removing all points below an objects surface
Blender has a built in tool for this called "Select side of Active". To use it you have to switch to vertex select mode and select exactly one vertex. This vertex marks the boundary, that means that either all vertices e.g. to its left or right will be selected after you run the command. See my screenshot for the result, the white vertex was the active one before i ran "Side of Active".

Using the last operator panel (Shortcut F6) you can tell blender which axis and which direction to look for vertices to select. Once done, simply delete them.